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\noindent
Eric   Rasmusen\footnote{ \begin{tiny}   Dan R. and Catherine M.
Dalton
Professor, Department of Business Economics and Public Policy, Kelley
School
of Business, Indiana University, Bloomington Indiana.   
\href{mailto:erasmuse@indiana.edu}{ Erasmuse@indiana.edu}.  \end{tiny}} \\
May 30, 2019  \hfill \url{http://www.rasmusen.org/papers/bargaining50.pdf}  \\

  

\noindent
 {\bf   Back to Bargaining Basics}
 
Players 1 and 2 are splitting a pie of size 1. Each simultaneously chooses a toughness level $x_i$ in $[0, \infty)$.  With probability $p(x_1, x_2)$, bargaining fails and each ends up with a payoff of zero. Otherwise, player 1 receives  $\pi(x_1, x_2)$ and Player 2 receives   $1-\pi(x_1, x_2)$. 


 \bigskip
  \noindent
{\bf Example 1: The Basics}. 
Let  $p(x_1, x_2) = \frac{x_1+x_2}{12}$ and  $\pi(x_1, x_2)=\frac{x_1 }{x_1+x_2}$. Equilibrium: $x_1=x_2=3$, $p = 50\%$. Each player's expected payoff is .25. 

  \bigskip
 \noindent
  {\bf General Model.  }
  Effort costs $c(x_i)$ for player $i$,  with $c \geq 0$,  $c' \geq 0, c'' \geq 0$.\\
   Utility is quasinlinear: $u_1(\pi) - c(x_1)$ and  $u_2(\pi) - c(x_2)$ with $u_1'>0, u_1'' \leq 0$ and $u_2'<0, u_2'' \leq 0$.\\
 Breakdown:    $p_1>0$  $p_2>0$, $p_{11} \geq 0$, $p_{22} \geq 0$,  and $p_{12} \geq 0$ for all values of $x_1, x_2$ such that $p<1$, and   $p_1= p_2=0$  for greater values.   $p(a,b) = p(b,a)$.\\
Player 1's share:  $\pi \in [0,1]$,   $\pi_1>0$,    $\pi_{11} \leq 0$, and   $\pi_{12} \geq 0$.   $\pi(a, b) = 1- \pi(b, a)$. \\
 These assumptions on $\pi$ imply that  $lim_{x_1 \rightarrow  \infty} \pi_1 \rightarrow 0$, since $\pi_1>0$, $\pi_{11} \leq 0$,  and $\pi \leq 1$.
  
\noindent
{\bf 
Proposition 1.}{\it The general model has a unique Nash equilibrium,   and that equilibrium is in pure strategies with a 50-50 split of the surplus: $x_1^*=x_2^*$ and   $\pi(x_1^*,x_2^*)=.5 $. }

  \bigskip
  \noindent
   {\bf Example 2:  A Vanishingly Small Probability of Breakdown. }
   Keep     $\pi(x_1, x_2)=\frac{x_1 }{x_1+x_2}$  but let  the breakdown probability be $p(x_1, x_2) = \frac{(x_1+x_2)^k}{12k^2}$  for $k$ to be chosen. In equilibrium, each player's expected payoff is close to  .5.
   
 \bigskip
 \noindent
   {\bf  N Players. }
  Player $i$'s payoff function  is 
$
 Payoff (i) =   (1- \frac{\sum_{i=1}^N x_i}{12}) \frac{x_i }{\sum_{i=1}^N x_i}.$    In equilibrium,  $\pi = 1/N$ and $  
 p(x, \dots,x)   =   \frac{(N-1)}{N }$.
 
 \bigskip
\noindent
{\bf Example 3: Unequal Bargaining Power}. 
Let the probability of breakdown be  $p(x_1, x_2) =Min\{ e^{(1-\theta) \beta x_1 +  \theta \beta x_2} -1, 1\}$, where $\theta \in [0,1]$ is player 1's bargaining power and $\beta>0$ is a parameter for breakdown risk.  Let  player 1's share of the pie be  $\pi(x_1, x_2)=.5 +   (x_1-x_2)$.  In equilibrium, player 1's share is $\theta$ and player 2's is $1-\theta$. 
 
   \bigskip
  \noindent
{\bf  Proposition 2:} {\it If   player 1 is more risk averse than player 2,  his   share   is smaller in equilibrium.   }

 \bigskip
  \noindent
       {\bf   Proposition 3. } {\it In the  multiperiod bargaining game, a  player's toughness and equilibrium share falls in his discount rate. }
         
 \bigskip
 \noindent
{\bf Example 6:  Player 1 Has an Outside Option of $z$}. As in Example 1, let the breakdown probability be 
  $p(x_1, x_2) = \frac{x_1+x_2}{12}$ and player 1's share   be  $\pi(x_1, x_2)=\frac{x_1 }{x_1+x_2}$.  Player 1 has an outside option of $z$, a payoff he receives if bargaining breaks down.  In equilibrium, player 1's share is $\frac{1}{2-z}$. 
 
  
  


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